Differentiatie equation

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February 6th 2009, 05:39 PM
elliotyang

Differentiatie equation

Suppose that the Earth were to stop moving tomorrow. It would of course immediately begin to fall towards the Sun. How long would we have before reaching the orbit of Venus [by which time we would all have been fried]? Information: the radius of the earth’s orbit is about 150
billion m, that of Venus’ orbit is about 100 billion m [both orbits being approximately circular],
the acceleration due to gravity at a distance of r from a central object of mass M is −GM/r^2,
where M [the mass of the Sun] is about 2 × 10^30 kg, and G is Newton’s constant = 6.67 × 10−11 in MKS units. So you have to solve the equation r''=-GM/r^2 (where r'' is differentiate with respect to t)

given by the equation, what i can get is
r'=sqrt[2GM(1/r-1/R) where R is the r at time t

how should i proceed after this?
the integral seem very tedious.
Can anyone show how to solve this in steps?
February 7th 2009, 04:08 AM
CaptainBlack

Quote:

Originally Posted by elliotyang

Suppose that the Earth were to stop moving tomorrow. It would of course immediately begin to fall towards the Sun. How long would we have before reaching the orbit of Venus [by which time we would all have been fried]? Information: the radius of the earth’s orbit is about 150
billion m, that of Venus’ orbit is about 100 billion m [both orbits being approximately circular],
the acceleration due to gravity at a distance of r from a central object of mass M is −GM/r^2,
where M [the mass of the Sun] is about 2 × 10^30 kg, and G is Newton’s constant = 6.67 × 10−11 in MKS units. So you have to solve the equation r''=-GM/r^2 (where r'' is differentiate with respect to t)

given by the equation, what i can get is
r'=sqrt[2GM(1/r-1/R) where R is the r at time t

how should i proceed after this?
the integral seem very tedious.
Can anyone show how to solve this in steps?

Here we have a linear motion problem, where:

$\frac{d^2r}{dt^2}=-\frac{GM}{r^2}$

with initial conditions $r(0)=R_{\oplus}$ , and $r'(0)=0$ .

Though you will probably find this easier to solve using conservation of energy.

CB
February 7th 2009, 04:52 AM
elliotyang

I got what you mean.
But how to integrate dr/sqrt[2GM(1/r-1/R)=dt where R=r(0)